Open Math. 2018; 16: 32–45 Open Mathematics Research Article Indu R. U. Churchill*, M. Elhamdadi, M. Green, and A. Makhlouf Ternary and n-ary f-distributive structures https://doi.org/10.1515/math-2018-0006 Received April 28, 2017; accepted January 10, 2018. Abstract: We introduce and study ternary f -distributive structures, Ternary f-quandles and more generally their higher n-ary analogues. A classication of ternary f -quandles is provided in low dimensions. More- over, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles. Furthermore, we give some computational examples. Keywords: Quandles, f-Distributive Structures, Ternary structures, Extensions, Cohomology MSC: 20G10, 17A30, 17A40 1 Introduction The rst instances of ternary operations appeared in the nineteenth century when Cayley considered cubic matrices. Ternary operations or more generally n-ary operations appeared naturally in various domains of theoretical and mathematical physics. The rst instances of ternary Lie algebras appeared in the Nambu’s Mechanics when generalizing hamiltonian mechanics by considering more than one hamiltonian [1]. The algebraic formulation of Nambu’s Mechanics was achieved by Takhtajan in [2]. Moreover, ternary algebraic structures appeared in String and Superstring theories when Basu and Harvey suggested to replace Lie algebra in the context of Nahm equations by a 3-Lie algebra. Furthermore, a ternary operation was used by Bagger-Lambert in the context of Bagger-Lambert-Gustavsson model of M2-branes and in the work of Okubo [3] on Yang-Baxter equation which gave impulse to signicant development on n-ary algebras. In recent years, there has been a growth of interests in many generalizations of binary structures to higher n-ary contexts. In Lie algebra theory, for example, the bracket is replaced by a n-ary bracket and the Jacobi identity is replaced by its higher analogue, see [4]. Generalizations of quandles to the ternary case were done recently in [5]. One may also mention reference [6] where the author uses two ternary operators, providing a generalization of a Dehn presentation which assigns a relation to each crossing in terms of the regions of the diagram that surround the crossing. For example, by coloring the four regions respectively a, b, c and d (see gure 2 in [6]), the author obtains d as a ternary function T(a, b, c) = ab−1c. This example of ternary operation was also considered in Example 2.8 in [5]. The author shows under certain conditions that ternary checkerboard colorings dene link invariants. The paper deals with dierent algebraic structures considered in [5] and in this work. In this paper we introduce and study a twisted version of ternary, respectively n-ary, generalizations of racks and quandles, where the structure is dened by a ternary operation and a linear map twisting the *Corresponding Author: Indu R. U. Churchill: Department of Mathematics, State University of New York at Oswego, Oswego, NY 13126, USA, E-mail: [email protected] M. Elhamdadi: Department of Mathematics, University of South Florida, Tampa, FL 33620, USA, E-mail: [email protected] M. Green: Department of Mathematics, University of South Florida, Tampa, FL 33620, USA, E-mail: [email protected] A. Makhlouf: Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, France, E-mail: [email protected] Open Access. © 2018 Churchill et. al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. Ternary and n-ary f -distributive structures Ë 33 distributive property. These type of algebraic structures, called sometimes Hom-algebras, appeared rst in quantum deformations of algebras of vector elds, motivated by physical aspects. A systematic study and mathematical aspects were provided for Lie type algebras by Hartwig-Larsson and Silvestrov in [7] whereas associative and other nonassociative algebras were discussed by the fourth author and Silvestrov in [8] and n-ary Hom-type algebras in [9]. The main feature of all these generalization is that the usual identities are twisted by a homomorphism. We introduce in this article the notions of ternary, respectively n-ary, f -shelf (resp. f-rack, f -quandle), give some key constructions and properties. Moreover, we provide a classication in low dimensions of f-quandles. We also study extensions and modules, as well as cohomology theory of these structures. For classical quandles theory, we refer to [10], see also [11–19]. For basics and some developments of Hom-type algebras, we refer to [8, 20–26]. This paper is organized as follows. In Section 2, we review the basics of f-quandles and ternary distribu- tive structures and give the general n-ary setting. In Section 3, we discuss a key construction introduced by Yau, we show that given a n-ary f-shelf (resp. f-rack, f-quandle) and a shelf morphism then one constructs a new n-ary f-shelf (resp. f-rack, f-quandle) and we provide examples. In Section 4, we provide a classication of ternary f -quandles in low dimensions. Section 5 gives the extension theory of f -quandles and modules. Finally, in Section 6 we introduce the cohomology of n-ary f-distributive structures and give examples. 2 f -quandles and ternary (resp. n-ary) distributive structures In this section we aim to introduce the notion of ternary and more generally n-ary f-quandles, generalizing the notion of f-quandle given in [27]. 2.1 A review of f -quandles and related structures First, we review the basics of the binary f-quandles. We refer to [27] for the complete study. Classical theory of quandle could be found in Denition 2.1. An f -shelf is a triple (X, *, f) in which X is a set, * is a binary operation on X, and f : X → X is a map such that, for any x, y, z 2 X, the identity (x * y) * f (z) = (x * z) * (y * z) (1) holds. An f -rack is an f-shelf such that, for any x, y 2 X, there exists a unique z 2 X such that z * y = f(x). (2) An f-quandle is an f-rack such that, for each x 2 X, the identity x * x = f(x) (3) holds. An f -crossed set is an f -quandle (X, *, f ) such that f : X → X satises x * y = f(x) whenever y * x = f (y) for any x, y 2 X. Denition 2.2. Let (X1, *1, f1) and (X2, *2, f2) be two f -racks (resp. f -quandles). A map ϕ : X1 → X2 is an f -rack (resp. f-quandle) morphism if it satises ϕ(a *1 b) = ϕ(a) *2 ϕ(b) and ϕ ◦ f1 = f2 ◦ ϕ. Remark 2.3. A category of f-quandles is a category whose objects are tuples (A, *, f) which are f -quandles and morphism are f -quandle morphisms. Examples of f-quandles include the following: 34 Ë I.R.U. Churchill et. al. – Given any set X and map f : X → X, then the operation x * y = f(x) for any x, y 2 X gives a f-quandle. We call this a trivial f -quandle structure on X. – For any group G and any group endomorphism f of G, the operation x * y = y−1xf (y) denes a f -quandle structure on G. – Consider the Dihedral quandle Rn, where n ≥ 2, and let f be an automorphism of Rn. Then f is given by f (x) = ax + b, for some invertible element a 2 Zn and some b 2 Zn [16]. The binary operation x * y = f (2y − x) = 2ay − ax + b (mod n) gives a f -quandle structure called the Dihedral f-quandle. – Any Z[T±1, S]-module M is a f-quandle with x * y = Tx + Sy, x, y 2 M with TS = ST and f (x) = (S + T)x, called an Alexander f-quandle. Remark 2.4. Axioms of Denition 2.1 give the following identity, (x * y) * (z * z) = (x * z) * (y * z). We note that the two medial terms in this equation are swapped (resembling the mediality condition of a quandle). Note also that the mediality in the general context may not be satised for f-quandles. For example one can check that the f-quandle given in item (2) of Examples is not medial. 2.2 Ternary and n-ary f -quandles Now we introduce and discuss the analogous notion of a f -quandle in the ternary setting and more generally in the n-ary setting. Denition 2.5. Let Q be a set, α be a morphism and T : Q×Q×Q → Q be a ternary operation on Q. The operation T is said to be right f-distributive with respect to α if it satises the following condition for all x, y, z, u, v 2 Q T(T(x, y, z), α(u), α(v)) = T(T(x, u, v), T(y, u, v), T(z, u, v)). (4) The previous condition is called right f -distributivity. Remark 2.6. Using the diagonal map D : Q → Q × Q × Q = Q×3 such that D(x) = (x, x, x), equation (4) can be written, as a map from Q×5 to Q, in the following form T ◦ (T × α × α) = T ◦ (T × T × T) ◦ ρ ◦ (id × id × id × D × D), (5) where id stands for the identity map. In the whole paper we denote by ρ : Q×9 → Q×9 the map dened as th th ρ = p6,8 ◦ p3,7 ◦ p2,4 where pi,j is the transposition i and j elements, i.e. ρ(x1, ··· , x9) = (x1, x4, x7, x2, x5, x8, x3, x6, x9).